What is the integral of (44-x)/(x^2+2x-24) ?

The integral of (44-x)/(x^2+2x-24) is

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integral of (44-x)/(x^2+2x-24)

\int \frac{44 - x}{x ^{2} + 2 x - 24} d x

- \frac{22}{5} ln \frac{1}{5} (x + 1) + 1 + \frac{22}{5} ln \frac{1}{5} (x + 1) - 1 - \frac{1}{2} ln x^{2} + 2 x - 24 - \frac{1}{10} (ln \frac{1}{5} (x + 1) + 1 - ln \frac{1}{5} (x + 1) - 1) + C

Solution steps

\int \frac{44 - x}{x ^{2} + 2 x - 24} d x

Expand

\frac{44 - x}{x ^{2} + 2 x - 24} : \frac{44}{x ^{2} + 2 x - 24} - \frac{x}{x ^{2} + 2 x - 24}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{44}{x ^{2} + 2 x - 24} d x - \int \frac{x}{x ^{2} + 2 x - 24} d x

\int \frac{44}{x ^{2} + 2 x - 24} d x = - \frac{22}{5} (ln \frac{x + 1}{5} + 1 - ln \frac{x + 1}{5} - 1)

\int \frac{x}{x ^{2} + 2 x - 24} d x = \frac{1}{2} ln x^{2} + 2 x - 24 + \frac{1}{10} (ln \frac{1}{5} (x + 1) + 1 - ln \frac{1}{5} (x + 1) - 1)

= - \frac{22}{5} (ln \frac{x + 1}{5} + 1 - ln \frac{x + 1}{5} - 1) - (\frac{1}{2} ln x^{2} + 2 x - 24 + \frac{1}{10} (ln \frac{1}{5} (x + 1) + 1 - ln \frac{1}{5} (x + 1) - 1))

Simplify

- \frac{22}{5} (ln \frac{x + 1}{5} + 1 - ln \frac{x + 1}{5} - 1) - (\frac{1}{2} ln x^{2} + 2 x - 24 + \frac{1}{10} (ln \frac{1}{5} (x + 1) + 1 - ln \frac{1}{5} (x + 1) - 1)) : - \frac{22}{5} ln \frac{1}{5} (x + 1) + 1 + \frac{22}{5} ln \frac{1}{5} (x + 1) - 1 - \frac{1}{2} ln x^{2} + 2 x - 24 - \frac{1}{10} (ln \frac{1}{5} (x + 1) + 1 - ln \frac{1}{5} (x + 1) - 1)

= - \frac{22}{5} ln \frac{1}{5} (x + 1) + 1 + \frac{22}{5} ln \frac{1}{5} (x + 1) - 1 - \frac{1}{2} ln x^{2} + 2 x - 24 - \frac{1}{10} (ln \frac{1}{5} (x + 1) + 1 - ln \frac{1}{5} (x + 1) - 1)

Add a constant to the solution

= - \frac{22}{5} ln \frac{1}{5} (x + 1) + 1 + \frac{22}{5} ln \frac{1}{5} (x + 1) - 1 - \frac{1}{2} ln x^{2} + 2 x - 24 - \frac{1}{10} (ln \frac{1}{5} (x + 1) + 1 - ln \frac{1}{5} (x + 1) - 1) + C

\int \frac{2 x ^{2} + 3 x + 2}{( x ^{2} + 1 ) ^{2}} d x

1 + e^{4 t}

x \to - 4 lim (8 x + 3)

y^{^{'}} + y = t e^{t}

x \frac{d y}{d x} + 4 y = 2 - \frac{7}{x}

What is the integral of (44-x)/(x^2+2x-24) ?
The integral of (44-x)/(x^2+2x-24) is